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Estimate an unmeasured midpoint value on a line graph · 8 practice problems

5.MD.B.2

Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 16 cm

The line graph shows the plant height recorded in a town. Estimate the value at about week 3.5, in $\text{cm}$ .

The graph is titled "Plant Height." The horizontal axis shows the week (1, 2, 3, 4, 5), and the vertical axis shows the value in $\text{cm}$ . The vertical axis is labeled 5, 10, 15; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,\text{cm}$ . The recorded values are: 1 $4\,\text{cm}$ , 2 $8\,\text{cm}$ , 3 $12\,\text{cm}$ , 4 $20\,\text{cm}$ , 5 $18\,\text{cm}$ . The value at week 3.5 was not recorded, so estimate it as the midpoint of the values at week 3 and week 4.

Show solution

Understand

A line graph of plant height shows 1 = 4, 2 = 8, 3 = 12, 4 = 20, 5 = 18 cm (each square = 1 cm). week 3.5 was not measured; I estimate it as the midpoint of the week 3 and week 4 readings.

Givens

1 = 4 cm, 2 = 8 cm, 3 = 12 cm, 4 = 20 cm, 5 = 18 cm
Each small grid square = 1 cm
week 3.5 is exactly halfway between week 3 and week 4
Estimate week 3.5 as the midpoint of the week 3 and week 4 values

Unknowns

The estimated value at about week 3.5 in cm

Constraints

week 3.5 lies between the two measured times, so its value should sit between 12 and 20

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. week 3.5 is the halfway time between week 3 and week 4, so I take the midpoint of 12 and 20.

Execute

#5 Look for a Pattern 5.MD.B.2

week 3.5 is halfway between week 3 (12) and week 4 (20). Add them: 12 + 20 = 32.

12 + 20 = 32

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 32 / 2 = 16 cm.

32 \div 2 = 16

On the graph 16 sits exactly halfway up the segment, matching week 3.5 halfway across.

Answer: 16 cm

Review

16 lies between 12 (week 3) and 20 (week 4), as it must for a time in between, and it is exactly in the middle, which fits week 3.5 being the middle time.

Use the rise (tool 8 / units): from week 3 to week 4 the value changed by 8, so half of that interval adds 8/2 to 12, giving 12 + 4 = 16 cm.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 2 answer: 30 people

The line graph shows the cafe customers recorded in a town. Estimate the value at about 10:30 a.m., in $\text{people}$ .

The graph is titled "Cafe Customers." The horizontal axis shows the hour (8, 9, 10, 11, 12), and the vertical axis shows the value in $\text{people}$ . The vertical axis is labeled 10, 20, 30; since 5 grid squares represent 10, each grid square represents $10 \div 5 = 2\,\text{people}$ . The recorded values are: 8 $12\,\text{people}$ , 9 $18\,\text{people}$ , 10 $24\,\text{people}$ , 11 $36\,\text{people}$ , 12 $30\,\text{people}$ . The value at 10:30 a.m. was not recorded, so estimate it as the midpoint of the values at 10 a.m. and 11 a.m..

Show solution

Understand

A line graph of cafe customers shows 8 = 12, 9 = 18, 10 = 24, 11 = 36, 12 = 30 people (each square = 2 people). 10:30 a.m. was not measured; I estimate it as the midpoint of the 10 a.m. and 11 a.m. readings.

Givens

8 = 12 people, 9 = 18 people, 10 = 24 people, 11 = 36 people, 12 = 30 people
Each small grid square = 2 people
10:30 a.m. is exactly halfway between 10 a.m. and 11 a.m.
Estimate 10:30 a.m. as the midpoint of the 10 a.m. and 11 a.m. values

Unknowns

The estimated value at about 10:30 a.m. in people

Constraints

10:30 a.m. lies between the two measured times, so its value should sit between 24 and 36

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. 10:30 a.m. is the halfway time between 10 a.m. and 11 a.m., so I take the midpoint of 24 and 36.

Execute

#5 Look for a Pattern 5.MD.B.2

10:30 a.m. is halfway between 10 a.m. (24) and 11 a.m. (36). Add them: 24 + 36 = 60.

24 + 36 = 60

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 60 / 2 = 30 people.

60 \div 2 = 30

On the graph 30 sits exactly halfway up the segment, matching 10:30 a.m. halfway across.

Answer: 30 people

Review

30 lies between 24 (10 a.m.) and 36 (11 a.m.), as it must for a time in between, and it is exactly in the middle, which fits 10:30 a.m. being the middle time.

Use the rise (tool 8 / units): from 10 a.m. to 11 a.m. the value changed by 12, so half of that interval adds 12/2 to 24, giving 24 + 6 = 30 people.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 3 answer: 13 degrees C

The line graph shows the town temperature recorded in a town. Estimate the value at about 1:00 p.m., in $^\circ\mathrm{C}$ .

The graph is titled "Town Temperature." The horizontal axis shows the time (8, 10, 12, 2, 4), and the vertical axis shows the value in $^\circ\mathrm{C}$ . The vertical axis is labeled 5, 10, 15; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,^\circ\mathrm{C}$ . The recorded values are: 8 $6\,^\circ\mathrm{C}$ , 10 $8\,^\circ\mathrm{C}$ , 12 $11\,^\circ\mathrm{C}$ , 2 $15\,^\circ\mathrm{C}$ , 4 $12\,^\circ\mathrm{C}$ . The value at 1:00 p.m. was not recorded, so estimate it as the midpoint of the values at 12 noon and 2 p.m..

Show solution

Understand

A line graph of town temperature shows 8 = 6, 10 = 8, 12 = 11, 2 = 15, 4 = 12 degrees C (each square = 1 degrees C). 1:00 p.m. was not measured; I estimate it as the midpoint of the 12 noon and 2 p.m. readings.

Givens

8 = 6 degrees C, 10 = 8 degrees C, 12 = 11 degrees C, 2 = 15 degrees C, 4 = 12 degrees C
Each small grid square = 1 degrees C
1:00 p.m. is exactly halfway between 12 noon and 2 p.m.
Estimate 1:00 p.m. as the midpoint of the 12 noon and 2 p.m. values

Unknowns

The estimated value at about 1:00 p.m. in degrees C

Constraints

1:00 p.m. lies between the two measured times, so its value should sit between 11 and 15

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. 1:00 p.m. is the halfway time between 12 noon and 2 p.m., so I take the midpoint of 11 and 15.

Execute

#5 Look for a Pattern 5.MD.B.2

1:00 p.m. is halfway between 12 noon (11) and 2 p.m. (15). Add them: 11 + 15 = 26.

11 + 15 = 26

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 26 / 2 = 13 degrees C.

26 \div 2 = 13

On the graph 13 sits exactly halfway up the segment, matching 1:00 p.m. halfway across.

Answer: 13 degrees C

Review

13 lies between 11 (12 noon) and 15 (2 p.m.), as it must for a time in between, and it is exactly in the middle, which fits 1:00 p.m. being the middle time.

Use the rise (tool 8 / units): from 12 noon to 2 p.m. the value changed by 4, so half of that interval adds 4/2 to 11, giving 11 + 2 = 13 degrees C.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 4 answer: 25 people

The line graph shows the library visitors recorded in a town. Estimate the value at about hour 3.5, in $\text{people}$ .

The graph is titled "Library Visitors." The horizontal axis shows the hour (1, 2, 3, 4, 5), and the vertical axis shows the value in $\text{people}$ . The vertical axis is labeled 10, 20, 30; since 5 grid squares represent 10, each grid square represents $10 \div 5 = 2\,\text{people}$ . The recorded values are: 1 $10\,\text{people}$ , 2 $16\,\text{people}$ , 3 $20\,\text{people}$ , 4 $30\,\text{people}$ , 5 $24\,\text{people}$ . The value at hour 3.5 was not recorded, so estimate it as the midpoint of the values at hour 3 and hour 4.

Show solution

Understand

A line graph of library visitors shows 1 = 10, 2 = 16, 3 = 20, 4 = 30, 5 = 24 people (each square = 2 people). hour 3.5 was not measured; I estimate it as the midpoint of the hour 3 and hour 4 readings.

Givens

1 = 10 people, 2 = 16 people, 3 = 20 people, 4 = 30 people, 5 = 24 people
Each small grid square = 2 people
hour 3.5 is exactly halfway between hour 3 and hour 4
Estimate hour 3.5 as the midpoint of the hour 3 and hour 4 values

Unknowns

The estimated value at about hour 3.5 in people

Constraints

hour 3.5 lies between the two measured times, so its value should sit between 20 and 30

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. hour 3.5 is the halfway time between hour 3 and hour 4, so I take the midpoint of 20 and 30.

Execute

#5 Look for a Pattern 5.MD.B.2

hour 3.5 is halfway between hour 3 (20) and hour 4 (30). Add them: 20 + 30 = 50.

20 + 30 = 50

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 50 / 2 = 25 people.

50 \div 2 = 25

On the graph 25 sits exactly halfway up the segment, matching hour 3.5 halfway across.

Answer: 25 people

Review

25 lies between 20 (hour 3) and 30 (hour 4), as it must for a time in between, and it is exactly in the middle, which fits hour 3.5 being the middle time.

Use the rise (tool 8 / units): from hour 3 to hour 4 the value changed by 10, so half of that interval adds 10/2 to 20, giving 20 + 5 = 25 people.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 5 answer: 13 degrees C

The line graph shows the garden temperature recorded in a town. Estimate the value at about 11 a.m., in $^\circ\mathrm{C}$ .

The graph is titled "Garden Temperature." The horizontal axis shows the time (6, 8, 10, 12, 2), and the vertical axis shows the value in $^\circ\mathrm{C}$ . The vertical axis is labeled 5, 10, 15; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,^\circ\mathrm{C}$ . The recorded values are: 6 $4\,^\circ\mathrm{C}$ , 8 $7\,^\circ\mathrm{C}$ , 10 $10\,^\circ\mathrm{C}$ , 12 $16\,^\circ\mathrm{C}$ , 2 $14\,^\circ\mathrm{C}$ . The value at 11 a.m. was not recorded, so estimate it as the midpoint of the values at 10 a.m. and 12 noon.

Show solution

Understand

A line graph of garden temperature shows 6 = 4, 8 = 7, 10 = 10, 12 = 16, 2 = 14 degrees C (each square = 1 degrees C). 11 a.m. was not measured; I estimate it as the midpoint of the 10 a.m. and 12 noon readings.

Givens

6 = 4 degrees C, 8 = 7 degrees C, 10 = 10 degrees C, 12 = 16 degrees C, 2 = 14 degrees C
Each small grid square = 1 degrees C
11 a.m. is exactly halfway between 10 a.m. and 12 noon
Estimate 11 a.m. as the midpoint of the 10 a.m. and 12 noon values

Unknowns

The estimated value at about 11 a.m. in degrees C

Constraints

11 a.m. lies between the two measured times, so its value should sit between 10 and 16

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. 11 a.m. is the halfway time between 10 a.m. and 12 noon, so I take the midpoint of 10 and 16.

Execute

#5 Look for a Pattern 5.MD.B.2

11 a.m. is halfway between 10 a.m. (10) and 12 noon (16). Add them: 10 + 16 = 26.

10 + 16 = 26

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 26 / 2 = 13 degrees C.

26 \div 2 = 13

On the graph 13 sits exactly halfway up the segment, matching 11 a.m. halfway across.

Answer: 13 degrees C

Review

13 lies between 10 (10 a.m.) and 16 (12 noon), as it must for a time in between, and it is exactly in the middle, which fits 11 a.m. being the middle time.

Use the rise (tool 8 / units): from 10 a.m. to 12 noon the value changed by 6, so half of that interval adds 6/2 to 10, giving 10 + 3 = 13 degrees C.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 6 answer: 18 cm

The line graph shows the snow depth recorded in a town. Estimate the value at about day 3.5, in $\text{cm}$ .

The graph is titled "Snow Depth." The horizontal axis shows the day (1, 2, 3, 4, 5), and the vertical axis shows the value in $\text{cm}$ . The vertical axis is labeled 5, 10, 15; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,\text{cm}$ . The recorded values are: 1 $6\,\text{cm}$ , 2 $10\,\text{cm}$ , 3 $14\,\text{cm}$ , 4 $22\,\text{cm}$ , 5 $18\,\text{cm}$ . The value at day 3.5 was not recorded, so estimate it as the midpoint of the values at day 3 and day 4.

Show solution

Understand

A line graph of snow depth shows 1 = 6, 2 = 10, 3 = 14, 4 = 22, 5 = 18 cm (each square = 1 cm). day 3.5 was not measured; I estimate it as the midpoint of the day 3 and day 4 readings.

Givens

1 = 6 cm, 2 = 10 cm, 3 = 14 cm, 4 = 22 cm, 5 = 18 cm
Each small grid square = 1 cm
day 3.5 is exactly halfway between day 3 and day 4
Estimate day 3.5 as the midpoint of the day 3 and day 4 values

Unknowns

The estimated value at about day 3.5 in cm

Constraints

day 3.5 lies between the two measured times, so its value should sit between 14 and 22

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. day 3.5 is the halfway time between day 3 and day 4, so I take the midpoint of 14 and 22.

Execute

#5 Look for a Pattern 5.MD.B.2

day 3.5 is halfway between day 3 (14) and day 4 (22). Add them: 14 + 22 = 36.

14 + 22 = 36

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 36 / 2 = 18 cm.

36 \div 2 = 18

On the graph 18 sits exactly halfway up the segment, matching day 3.5 halfway across.

Answer: 18 cm

Review

18 lies between 14 (day 3) and 22 (day 4), as it must for a time in between, and it is exactly in the middle, which fits day 3.5 being the middle time.

Use the rise (tool 8 / units): from day 3 to day 4 the value changed by 8, so half of that interval adds 8/2 to 14, giving 14 + 4 = 18 cm.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 7 answer: 24 degrees C

The line graph shows the pool temperature recorded in a town. Estimate the value at about 11:30 a.m., in $^\circ\mathrm{C}$ .

The graph is titled "Pool Temperature." The horizontal axis shows the time (9, 10, 11, 12, 1), and the vertical axis shows the value in $^\circ\mathrm{C}$ . The vertical axis is labeled 15, 20, 25; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,^\circ\mathrm{C}$ . The recorded values are: 9 $18\,^\circ\mathrm{C}$ , 10 $20\,^\circ\mathrm{C}$ , 11 $22\,^\circ\mathrm{C}$ , 12 $26\,^\circ\mathrm{C}$ , 1 $28\,^\circ\mathrm{C}$ . The value at 11:30 a.m. was not recorded, so estimate it as the midpoint of the values at 11 a.m. and 12 noon.

Show solution

Understand

A line graph of pool temperature shows 9 = 18, 10 = 20, 11 = 22, 12 = 26, 1 = 28 degrees C (each square = 1 degrees C). 11:30 a.m. was not measured; I estimate it as the midpoint of the 11 a.m. and 12 noon readings.

Givens

9 = 18 degrees C, 10 = 20 degrees C, 11 = 22 degrees C, 12 = 26 degrees C, 1 = 28 degrees C
Each small grid square = 1 degrees C
11:30 a.m. is exactly halfway between 11 a.m. and 12 noon
Estimate 11:30 a.m. as the midpoint of the 11 a.m. and 12 noon values

Unknowns

The estimated value at about 11:30 a.m. in degrees C

Constraints

11:30 a.m. lies between the two measured times, so its value should sit between 22 and 26

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. 11:30 a.m. is the halfway time between 11 a.m. and 12 noon, so I take the midpoint of 22 and 26.

Execute

#5 Look for a Pattern 5.MD.B.2

11:30 a.m. is halfway between 11 a.m. (22) and 12 noon (26). Add them: 22 + 26 = 48.

22 + 26 = 48

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 48 / 2 = 24 degrees C.

48 \div 2 = 24

On the graph 24 sits exactly halfway up the segment, matching 11:30 a.m. halfway across.

Answer: 24 degrees C

Review

24 lies between 22 (11 a.m.) and 26 (12 noon), as it must for a time in between, and it is exactly in the middle, which fits 11:30 a.m. being the middle time.

Use the rise (tool 8 / units): from 11 a.m. to 12 noon the value changed by 4, so half of that interval adds 4/2 to 22, giving 22 + 2 = 24 degrees C.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!

Variant 8 answer: 13 degrees C

The line graph shows the road temperature recorded in a town. Estimate the value at about 12 noon, in $^\circ\mathrm{C}$ .

The graph is titled "Road Temperature." The horizontal axis shows the time (7, 9, 11, 1, 3), and the vertical axis shows the value in $^\circ\mathrm{C}$ . The vertical axis is labeled 5, 10, 15; since 5 grid squares represent 5, each grid square represents $5 \div 5 = 1\,^\circ\mathrm{C}$ . The recorded values are: 7 $3\,^\circ\mathrm{C}$ , 9 $6\,^\circ\mathrm{C}$ , 11 $9\,^\circ\mathrm{C}$ , 1 $17\,^\circ\mathrm{C}$ , 3 $14\,^\circ\mathrm{C}$ . The value at 12 noon was not recorded, so estimate it as the midpoint of the values at 11 a.m. and 1 p.m..

Show solution

Understand

A line graph of road temperature shows 7 = 3, 9 = 6, 11 = 9, 1 = 17, 3 = 14 degrees C (each square = 1 degrees C). 12 noon was not measured; I estimate it as the midpoint of the 11 a.m. and 1 p.m. readings.

Givens

7 = 3 degrees C, 9 = 6 degrees C, 11 = 9 degrees C, 1 = 17 degrees C, 3 = 14 degrees C
Each small grid square = 1 degrees C
12 noon is exactly halfway between 11 a.m. and 1 p.m.
Estimate 12 noon as the midpoint of the 11 a.m. and 1 p.m. values

Unknowns

The estimated value at about 12 noon in degrees C

Constraints

12 noon lies between the two measured times, so its value should sit between 9 and 17

Plan

#5 Look for a Pattern · also uses: #1 Draw a Diagram

On a line graph the segment between two points rises steadily, so the value halfway across is the average of the two endpoints. 12 noon is the halfway time between 11 a.m. and 1 p.m., so I take the midpoint of 9 and 17.

Execute

#5 Look for a Pattern 5.MD.B.2

12 noon is halfway between 11 a.m. (9) and 1 p.m. (17). Add them: 9 + 17 = 26.

9 + 17 = 26

The connecting line climbs evenly, so the middle is the average of the ends.

#1 Draw a Diagram 5.MD.B.2

Divide the sum by 2 to find the midpoint value: 26 / 2 = 13 degrees C.

26 \div 2 = 13

On the graph 13 sits exactly halfway up the segment, matching 12 noon halfway across.

Answer: 13 degrees C

Review

13 lies between 9 (11 a.m.) and 17 (1 p.m.), as it must for a time in between, and it is exactly in the middle, which fits 12 noon being the middle time.

Use the rise (tool 8 / units): from 11 a.m. to 1 p.m. the value changed by 8, so half of that interval adds 8/2 to 9, giving 9 + 4 = 13 degrees C.

Standards · min grade 5

5.MD.B.2 Make a line plot to display a data set and solve problems using the data — Reading the two endpoint values and estimating the in-between point

💡 Halfway across a line segment means halfway up - just average the two end values!